MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  reupick3 Structured version   Visualization version   GIF version

Theorem reupick3 4210
Description: Restricted uniqueness "picks" a member of a subclass. (Contributed by Mario Carneiro, 19-Nov-2016.)
Assertion
Ref Expression
reupick3 ((∃!𝑥𝐴 𝜑 ∧ ∃𝑥𝐴 (𝜑𝜓) ∧ 𝑥𝐴) → (𝜑𝜓))
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)

Proof of Theorem reupick3
StepHypRef Expression
1 df-reu 3111 . . . 4 (∃!𝑥𝐴 𝜑 ↔ ∃!𝑥(𝑥𝐴𝜑))
2 df-rex 3110 . . . . 5 (∃𝑥𝐴 (𝜑𝜓) ↔ ∃𝑥(𝑥𝐴 ∧ (𝜑𝜓)))
3 anass 469 . . . . . 6 (((𝑥𝐴𝜑) ∧ 𝜓) ↔ (𝑥𝐴 ∧ (𝜑𝜓)))
43exbii 1830 . . . . 5 (∃𝑥((𝑥𝐴𝜑) ∧ 𝜓) ↔ ∃𝑥(𝑥𝐴 ∧ (𝜑𝜓)))
52, 4bitr4i 279 . . . 4 (∃𝑥𝐴 (𝜑𝜓) ↔ ∃𝑥((𝑥𝐴𝜑) ∧ 𝜓))
6 eupick 2687 . . . 4 ((∃!𝑥(𝑥𝐴𝜑) ∧ ∃𝑥((𝑥𝐴𝜑) ∧ 𝜓)) → ((𝑥𝐴𝜑) → 𝜓))
71, 5, 6syl2anb 597 . . 3 ((∃!𝑥𝐴 𝜑 ∧ ∃𝑥𝐴 (𝜑𝜓)) → ((𝑥𝐴𝜑) → 𝜓))
87expd 416 . 2 ((∃!𝑥𝐴 𝜑 ∧ ∃𝑥𝐴 (𝜑𝜓)) → (𝑥𝐴 → (𝜑𝜓)))
983impia 1110 1 ((∃!𝑥𝐴 𝜑 ∧ ∃𝑥𝐴 (𝜑𝜓) ∧ 𝑥𝐴) → (𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1080  wex 1762  wcel 2080  ∃!weu 2610  wrex 3105  ∃!wreu 3106
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1778  ax-4 1792  ax-5 1889  ax-6 1948  ax-7 1993  ax-10 2111  ax-12 2140
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-ex 1763  df-nf 1767  df-sb 2042  df-mo 2575  df-eu 2611  df-rex 3110  df-reu 3111
This theorem is referenced by:  reupick2  4211  fvineqsneq  34237
  Copyright terms: Public domain W3C validator